When carbon dioxide is passed through limewater, you see a milky white precipitate, CaCO_3 (s). Lime water is saturated Ca(OH)_2(aq); Ca(OH)_2 is sparingly soluble in water. CaCO_3, alike with most carbonates, is pretty insoluble stuff; much more so that Ca(OH)_2.
Its reaction with CO_2 (g) may be represented:
Ca(OH)_2(aq) + CO_2(g) rarr CaCO_3(s)darr + H_2O(l)
Note that this equation is balanced.
If you keep bubbling CO_2, the white solid should go up. You should form the bicarbonate, Ca(HCO_3)_2, which is soluble in water, and the reaction is outside the A level syllabus. As a further tip, the best source of carbon dioxide outside of a cylinder, is carbonated water (fizzy water; water supersaturated with CO_2(g)); you can cheaply pick up litres of this in a supermarket.
None of what I have written here is a substitute for doing the reaction in a laboratory
